Estimating Tail Development Factors: What to do When
the Triangle Runs Out
Joseph Boor, FCAS
Abstract:: There are several methods in use today for estimating tail factors. However,
most of them are discussed as adjuncts to papers that primarily deal with other subjects.
This paper will present a wide variety of method in an understandable format, and includes
copious examples
Keywords: Loss resenting, tail factors
1. INTRODUCTION
In ma W loss reserve analyses, especially those involving long-tail casualty lines, the loss
development mangle may end before all the claims are settled and before the final costs of
any year are known. For example, it is quite common to analyze U. S. workers
compensation loss reserve needs using the ten years of data available in Schedule P of the
US NAIC-mandated Annual Statement, while knowing that some of the underlying claims
may take as long as fifty years to close. In response to this, actuaries supplement the 'link
ratios' they obtain from the available mangle data with a 'tail factor' that estimates the
development beyond the last stage of development (last number of months of maturity,
usually) for which a link ratio could be calculated.
The tail factor is used just like a link ratio in that it estimates (1.0 + ratio of (final costs after
all daims are closed) to (the costs as of the last development stage used)). It is of course
included in the product of all the remaining link ratios beyond any given stage of
development in calculating a loss development factor to ultimate for that stage of
development.
This paper will discuss the methods of computing (really estimating to be precise) tail factors
in common usage today. It will also suggest both improvements in existing methods and a
new method. It will begin with the simplest class of methods and move forward in
increasing complexity.
There are four groups of methods that will be presented:
1. The Bondy (repeat-the-last link-type) methods
2. The Algebraic methods (methods based on algebraic relationships between the paid
and incurred mangles)
3. Use of Benchmark Data
4. Curve Fitting Methods
Casualty Actuarial Society Forum, Winter 2006
345
Estimating Tail DevelopmentFactors
As part of the discussion, commentary on the advantages and disadvantages of each
individual method, as well as each class of methods will be included. When the opporttmity
to discuss an improvement or enhancement that applies to multiple methods presents itself,
a brief digression on the enhancement will be included.
2. G R O U P 1 - THE BONDY-TYPE METHODS
The Bondy methods all arose from an approach published by Martin Bondy prior to the
1980s. In what was thought to be a period where development decayed rapidly from link
ratio to link ratio, he promulgated a practice of simply repeating the last link ratio for use as
the tail factor. Since then, several variations of his method that all base the tail factor on the
last available link ratio have arisen.
2.1 The Bondy Method
As explained above, the original Bondy method involves simply using the last link ratio that
could be estimated from the triangle (the link ratio of the last development stage present in
the triangle, or the last stage where the triangle data could be deemed reliable for estimation)
as the tail factor. This 'repeat the last link ratio' approach probably seems crude and
unreasonable for long-tailed lines, where link ratios decay slowly. However, for fast deca3qng
lines (such as an accident year1 analysis of automobile extended warranty) this method may
work when used as early as thirty-six or forty-eight months of maturity. It must be
recognized, though, that in long-tailed lines the criticism is usually justified.
To truly understand this method it also may be best viewed in historical context. The author
of the method, Martin Bondy, developed this method well prior to the 1980's. It is
commonly believed that during the 1960s and certainly part of the 1970s the courts
proceeded at a faster pace and, ignoring the long-taB asbestos, environmental, and mass tort
issues that would eventually emerge, general liability was believed to have a much shorter tail
than we see today.
It is also of interest to note that there is a theoretical foundation that supports this in certain
circumstances. If one assumes that the 'development portion' of the link ratios (the link
ratios minus one) are decreasing by one-half at each stage of development, and the last link
ratio is fairly low, then the theoretically correct tail factor to follow a link ratio of l + d is:
(1 +.58) x (1 + .25d) x (1 +. 125d) x (1 +.0625d) × .......
Or
1+(.5+.25+.125+.0625+....)×d + terms revolving d 2, d ~, etc.
It should be noted that policy year automobile extended warranty represents an entirely different situation.
346
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
Which, per the interest theorem v+ve+v3+ .... =v/(1-v) is equivalent to:
1+1 xd + terms involving d z, d 3, etc.
Since d is 'small', the other terms will be smaller by an order of magnitude, making the
implied tail factor under these assumptions very close to the Bondy tail factor, a repetition of
the last link ratio, l+d. So the Bondy tail factor is 'nearly' equivalent to the tail implied by
what will later be called the 'exponential decay' method, with a 50% decay constant.
Of note, this involves two basic assumptions. First that the link ratios decay in proportion
to the remaining 'development portion' of the link. O f note, in the absence of any
information whatsoever about the decay, that would be as reasonable an assumption as one
could reasonably make. Second, that the decay constant is 50%. Again, in the absence of any
data whatsoever, one-half would be as reasonable an assumption as one could possibly
make. Of course, we do have data in the link ratios before the tail, but it is important to
understand this theoretical basis for the Bondy tail factor.
2.2 The Modified Bondy Method
In this method, the last link ratio available from the triangle, call it l+d, is modified by
multiplying the development portion by 2. The result is a development factor like l+2d.
Alternately, the last entire link ratio may be squared, which yields nearly the same value.
This has many of the same issues and applications as the basic Bondy method, but it does
field a larger tail than the Bondy method itself. However, for long-taft lines it is still not
what would be considered a truly conservative approach, as we will see later. The
assumption here is ' The Bondy method seems to underestimate, it should be increased, the
easiest thing to do is to multiply the development portion by two.'.
A little algebra and the v+v2+v3+ .... =v/(1-v) theorem show that this is functionally
equivalent to 'exponential decay' with a decay coefficient of 2/3.
2.3 Advantages and Disadvantages of the Bondy Methods
The primary advantages of the ]3ondy methods are that they are extremely simple to execute
and easy to understand. Further, they involve relatively straightforward assumptions.
However, a major disadvantage is that they tend to greatly underestimate tabs of long-tailed,
slow-decaying lines.
3. GROUP 2 - THE ALGEBRAIC METHODS
These methods involve initially computing some algebraic quantity that in turn describes a
relationship between some aspect of the paid and incurred loss triangles. Then that quantity
can be used to generate a tail factor estimate. As with the Bondy method, and almost all tail
factor estimation methods, they are based on assumptions. However, in this case each is
based on some relatively simple and fairly logical assumption that some numerical
relationship known to be true in one circumstance will be true in another.
Casualty Actuarial Society Forum,Winter 2006
347
Estimating Tail DevelopmentFactors
3.1 Equalizing Paid and Incurred Development Ultimate Losses
This method is the first method discussed with a full theoretical background. It is most
useful when incurred loss development essentially stops after a certain stage (i.e., the link
ratios are near to unity or unity). Then, due to the absence of continuing development, the
current case incurred (sometimes called reported) losses are a good predictor of the ultimate
losses for the older or oldest years without a need for additional tail factor development. A
tail factor suitable for paid loss development can then be computed as the ratio of the case
incurred losses to-date for the oldest (accident-') year in the triangle divided by the paid losses
to-date for the same (accident) year. That way, the paid and incurred development tests will
produce exactly the same ultimate losses for that oldest year.
This method relies on one axiomatic (meaning plainly true rather than an assumption as
such) assumption and two true assumptions. The axiomatic assumption is that the paid loss
and incurred loss development estimates of incurred loss are estimating the same quantity,
therefore the ultimate loss estimates they produce should be equal. The second assumption
(the first true assumption) is that the incurred loss estimate of the ultimate losses for the
oldest year is accurate. The last assumption is that the other years will show the same
development in the tail as the oldest year.
This method may also be generalized to the case where case incurred losses are still showing
development near the tail. In that case, the implied paid loss tail factor is
(incurred loss development ultimate loss estimate for the oldest year) /
date for the oldest year).
(paid losses to-
O f course, in that instance the incurred loss development estimate for the oldest (accident)
year is usually the case incurred losses for the oldest year multiplied by an incurred loss tail
factor developed using other methods.
This method has a substantial advantage in that it is based solely on the information in the
triangle itself and needs no special assumptions. Its weakness is that you must already have a
reliable estimate of the ultimate loss for the oldest year before it can be used. An ancillary
weakness flows from the assumptions underlying this method. Specifically, if the initial
incurred loss development test is driven by a tail factor assumption, this becomes a test that
is also based on not only that assumption, but also the assumption that the ratio of the case
incurred loss to the paid loss WIU be the same for the less mature years once they reach the
older level of maturity where you are equalizing the paid and incurred loss estimates.
2 Accident year is used here for illustration. Under similar circumstances, this method would also work in
policy year, reinsurance contract year, etc. development.
348
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
3.1.1 An example:
Assume that it is just after year-end of 2000. You have pulled the incurred loss triangle from
a carrier by subtracting part 4 of Schedule P from part 2 of Schedule P. You have also
pulled a paid loss triangle from part 3 of Schedule P. The mangles cover 1991-2000, so 1991
is the oldest year. Say for the sake of argument that the incurred loss link ratios you develop
are 2.0 for 12-24 months, 1.5 for 24-36, 1.25 for 36-48, 1.125 for 48-60, 1.063 for 60-72,
1.031 for 72-84, 1.016 for 84-96, 1.008 for 96-108, and 1.004 for 108-120. This conveniently
happens to match the exponential decay discussed for the Bondy method, so it makes sense
to use 1.004 for the tail factor for development beyond 120 months. Now assume that the
latest available (i.e., at 12/31/2000, or 120 months maturity) the case incurred loss 3 for 1991
is $50,000,000 and the corresponding paid loss is $40,000,000. The incurred test ulfrnate
using the 1.004 tail factor is $50,200,000. The paid loss tail factor to equalize the ulfmate
would be $50~200,000 divided by $40,000,000 or 1.255.
3.1.2 I m p r o v e m e n t 1 - using multiple years tO develop the tail factor
As stated earlier, the previous method assumes that the current ratio of case incurred loss to
paid loss that exists in the oldest year (1991 evaluated at 12/31/2000 in the example above)
will apply to the other years when they reach that same level of maturitT. For a large high
dollar volume mangle with relatively low underlying pohcy limits that may be a reasonable
assumption, but for many reserving applications the 120 month ratio of case incurred to paid
loss may depend on whether a few large, complex claims remain open or not. Therefore, it
may be wise to supplement the tail factor derived from the oldest available year with that
implied by the following year or even the second following year. This method is particularly
useful when the later development portion of the mangle has some credibility, but the
individual link ratio estimates from the development triangle are not fully credible.
The process of doing so is fairly straightforward. You merely compute the tail factor for
each succeeding year by the method above, and divide each by the remaining link ratios in
the mangle.
An example using the data above may help clarify matters. Given the data above, assume
that 1992 has $50,000,000 of paid loss and $60,000,000 of case incurred loss. Also, assume
that your best estimate of the 108-120 paid loss link ratio is 1.01. The incurred loss esfmate
of the ultimate loss, using the 108-120 link ratio (1.004) and the incurred loss tail factor (also
1.004) is $60,000,000×1.004×1.004, or $60,480,960. The estimated (per incurred loss
development) ultimate loss to paid loss ratio at 108 months would then be
$60,480,960/$50,000,000, or approximately 1.210. So, 1.210 would then be the tail factor
estimate for 108 months. Dividing out the 108-120 paid link ratio (assumed above to be
1.01) gives a tail factor for 120 months of 1.21/1.01 = 1.198. By comparison, the previous
analysis using 1991 instead of 1992 gave a 120-monfll tail factor estimate of 1.255. So it is
possible that either 1991 has a high number of claims remaining open, or that 1992 has a low
number. Both indicate taft factors in the 120-125 approximate range, though. So averaging
3 To be technicallycorrect, this would be loss and defense and cost containment under 2003 accounting
rules.
Casualty Actuarial Society Forum, Winter 2006
349
Estimating Tail DevelopmentFactors
the estimates might be prudent. Further, the use of averaging greatly limits the impact of
any unusually low or high case reserves that may be present in the oldest year in the mangle,
Note also, that the improvement above involved computing an alternate tail factor using the
year with one year less maturity. A similar analysis could also be performed on the next
oldest year, 1993, except that two incurred development link ratios plus the tail factor are
needed to compute the incurred loss estimate of ultimate. Correspondingly, two paid loss
link ratios need to be divided out of the (incurred loss ultimate estimate)/(paid loss to-date
ratio for 1993) to estimate the 120 month paid loss tail factor
3.1.2.1 An important note
Further, in this case the improvement involved reviewing the taft factors at various ages from
the equalization of paid and incurred loss estimates of the ultimate loss. The core process
involves computing tail factors at different mamries, then dividing by the remaining link
ratios to place them all at the same maturity. As such, it can also be used in the context of
other methods for computing tail factors that will be discussed later in this paper.
3.1.3 A brief digression - the nrimarv activity w i t h i n e a c h d e v e l o p m e n t sta~e
v
When using multiple years to estimate a tail factor, it is relatively important that the years
reflect the same general type of claims department activity as that which takes place in the
tail. For example, in the early 12 to 24 month stage of workers compensation, the primary
development activity is the initial reporting of claims and the settlement and closure of small
claims. The primary factors influencing development are how quickly the claims are
reported and entered into the system, and the average reserves (assuming the claims
department initially just sets a 'formula reserve', or a fixed reserve amount for each claim of
a given type such as medical or lost time) used when claims are first reported. In the 24 to
36-48 month period, claims department activity is focused on ascertaining the true value of
long-term claims and settling medium-sized claims. After 48-60 months most of the activity
centers on long-term claims. So, the 12-24 link ratio has relatively httle relevance for the tail,
as the driver behind the link ratio is reporting and the size of initial formula reserves rather
than the handling of long-term cases. Similarly, if the last credible link ratio in the mangle is
the 24 to 36 or 36 to 48 link ratio, that mangle may be a poor predictor of the required tail
factor.
350
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
3.2 The Sherman-Boor4 Method - Adjusting the Ending Case Using
Ratios of Paid Loss to Case Reserve Disposed Of
This method, developed by Sherman in Section X of [3] and independently by the author, is
the one method that relies solely on the triangle itself and does not require a pre-existing
ultimate loss estinaate, involve curve-fitting assumptions, or require external data. For data
triangles with high statistical reliability as predictors, this can represent the optimum
estimation process.
This method involves simply determining the ratio of case reserves to paid loss for the oldest
year in the triangle, then adjusting the case reserves by an estimate o f the ratio of the unpaid
loss to carried case reserves. In essence, the case reserves of the oldest accident year are
'grossed up' to estimate the true unpaid loss using a factor. The estimate of the (true unpaid
loss)/(case reserves) factor is based on how many dollars o f payments are required to
'eliminate' one dollar of case reserves.
The mathematical formula requires computing a mangle containing incremental rather than
cumulative paid losses. In effect, for each point in the paid loss triangle, one need only
subtract the prexdous value in the same row (the first column is o f course unchanged). The
next step begins with a triangle o f case reserves. The incremental case reserve disposed of is
calculated as the case reserve in the same row before the data point, less the current case
reserve. That represents (as the beginning case reserve - the ending case reserve) the case
reserve disposed of. Then the ratios of incremental paid to reserve disposed of at the same
points in the triangles are computed. Reviewing these, the adjustment ratio for the ending
case reserves is estimated.
3.2.1 A n e x a m p l e
Reviewing an example may help the reader follow the calculations discussed earlier. This
method requires two triangles, one of paid loss and one of case reserves. Consider the
following set of triangles:
Cumulative Paid Loss Triansle
12
1991
1,000
1992
1,100
1993
1,300
1994
1,200
1995
1,400
1996
1,490
24
2,000
2,400
2,500
2,300
2,800
36
2,500
3,000
3,000
3,100
4{
2,800
3,50(
3,400
6C
2,95C
3,90C
72
3,100
4 Of note, this method was first published by Richard Sherman, FCAS in 1984 and developed
independently by the author in 1987. Of note, the author used some business materials that contained
precursors to this method in 1984-1986 that were developed by a firm of which Mr. Sherman was a
principal.
Casualty Actuarial Society Forum, Winter 2006
351
Estimating Tail DevelopmentFactors
Triangle ofCase Reserves Outstandin~ (Cumulanve Case Incurred-Cumuladve Pmd)
12
24
36
4~
1991]
1,500
1,30(
900
75C
1992[
2,000
1,70(
1,300
90C
1993]
1994[
1995]
1996]
1,900
2,100
2,300
2,500
1,70(
2,10(
2,00(
1,300
1,500
60
600
600
72]
5001
1,00C
First, we c o m p u t e the incremental paid loss triangle. We begin with a given cell in the
cumulative paid loss triangle, and then we subtract the previous cell in the same row o f the
cumulative paid loss triangle. T h a t produces the following triangle.
Incremental P~dLossTfian#e
12
199'
1,00C
1992
1,10C
1993
1,30C
1994
1,20C
1995
1,40C
1996
1,49C
24
1,000
1,300
1,200
1,100
1,400
36
50C
600
500
800
48
300
500
400
60
150
400
72!
I
150I
T h e n we subtract the current cell from the previous cell in the case reserve triangle to obtain
the triangle o f case reserves disposed of.
Triangle oflncremental Case Resen,esDisposedOf
12
24
1991
1992
1993
1994
1995
1996
352
20(
30(
20(
10(
30(
36
48
60
72
40C
40G
40G
60{3
150
400
300
150
300
100
Casualty Actuarial Society Forum,Winter 2006
Estimating Tail DevelopmentFactors
Then we divide the actual fmal costs paid (the incremental paid loss), by the assumptionbased case reserves eliminated.
Ratio ofPmdLossto Resen,es Eliminated
Ii
24
36
48
6£
72
200%
125%
133%
100%
133%
150~
1991
1992
1993
500~
433~
600~
125%
150~
125~
1994
1995
1996
1100%
467~
133%
Because the early development invoR-es not just elimination of case reserves through
payments, but also substantial emergence of IBNR claims, the 12 and 36 columns are
presumably distorted. In many lines the 48 month column would still be heavily affected by
newly reported large claims, but presumably this is medium-tail business. Looking at the
various ratios it would appear that they average around 140%, so we will use that as our
adjustment factor for the case reserves.
Pulling the $500 of case left on the 1991 year at 72 months, and the cumulative paid on the
1991 year of $3,100, the development portion of the paid loss tail factor would be
($500/$3,100)x 140% = .161x140% = .226. So, the paid loss tail factor would be 1.226.
For the incurred loss tail factor, first note that only the 'development portion' of the 140%,
or 40%, need be applied (the remaining case is already contained in the incurred). Second, a
ratio of the case reserves to incurred loss is technically needed (replacing 1.61 with
$500/($500+$3,100) = .139). Multiplying the two numbers creates an estimate of the
development portion of the tail at .4x.139=.056. So, the incurred loss tail factor estimate
would be 1.056.
3.2.2 An Important N o t e
As is the case with most of the other methods, this method has strengths and weaknesses.
Significant strengths o f this method are that it requires only the data already in the triangle
and that it does not require additional assumptions. The weakness is that it can be distorted
if the adequacy of the ending case has changed significantly from the previous year. The
reader is advised to also follow Improvement 1 and also evaluate the tail at the next-to-oldest
year.
Casualty Actuarial Society Forum,Winter 2006
353
Estimating Tail DevelopmentFactors
4. GROUP THREE- METHODS THAT USE BENCHMARK DATA
A common solution to the ratemaking problems generated by data with partial statistical
rehability (credibility) is to supplement the claims data with a 'complement of credibility'. O f
course, tail factor estimation problems stem more from a lack of any data at all after the
oldest development stage in the triangle rather than from partially rehable data. But, we can
adopt a similar strategy and add outside data in the form of benchmark development factors.
4.1 Directly Using Tail Factors From Bencbmark Data
As noted above, many actuaries review benchmark data in selecting tail factors s. Benchmark
data may come from one of several sources. Perhaps the most common is the use of the
data triangles that can be developed from Best's Aggregates and Averages for each of the
Schedule P lines. The two larger rating bureaus, the National Council on Compensation
Insurance and Insurance Services Office; as well as the Reinsurance Association of America,
all publish benchmark loss development data. At its simplest, this method invokes cop3fing
the derived remaining development factor at the maturity desired for the tail factor.
It is important to note, though, that the quality of the benchmark tail factor as an estimate of
the tail depends on how closely the tail development of the benchmark mirrors the tail
development of the book of business being analyzed. Considerations such as differences in
the way claims are adjusted or reserved, differences in the potential for long-developing high
value claims, differences in the initial reporting pattern of claims (claims-made vs.
occurrence, whether or there is an innately long discovery period or not, etc.), and
differences in the adjudication process of litigated claims can all cause differences in
development patterns. It is important to consider those factors along with the statistical
reliability of the benchmark triangle when selecting the most appropriate benchmark tail
factor.
4.2 Using Bencbmark Tail Factors Adjusted to Company Development
Levels
One way to address differences between the benchmark development pattern and the
development pattern of a given book of business is to try to adjust the benchmark data to
more closely mirror the subject book of business. A common practice is to review the
relafivities of link ratios from the triangle being analyzed to benchmark link ratios. O f
course, there is not a tail factor for the triangle being analyzed (we are Wing to estimate
one). So, instead we can review the quotients (relativities) of subject triangle link ratios to
those of the benchmark data at the development stages prior to the tail development stage.
The relativities from those stages are used to estimate a adjustment multiplier for the
benchmark tail factor. O f note, generally just the development portions ('d' of l+d) are
compared in all the relativities we compute.
s It is also common for actuaries to review benchmark data to supplement the portion of the reserve triangle
following 72, 60, 48, or even 36 months when the overall triangle has medium credibility and hence has
less than medium credibility in the portion that is dominated by activity on a smallish number of claims.
354
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
4.2.1 An examnle
A n example will help to illustrate how the process works.
patterns:
Consider the following two
Link
Ratio
Estimated Benchmark
Months of
By
Link
Maturity Triangle
Ratio
12
2.00(
2.00(
24
1.45(
1.35(
36
1.20(
1.15(
48
1.15(
1.10(
6C
1.10(
1.05(
72
1.08(
1.03(
84
1,05(
1.025
9~
1.035
1.02(
108
1,01(
1,01(
Ta/
1.05(
We then simply compute the relativity quotient of the 'development portion' o f our trianglebased link ratios to the development portion o f the matching benchmark link ratios. Noting
that 1+1 = 100%, .45+.35 = 129%, .2+.15 = 133%, etc.
Casualty Actuarial Society Forum,Winter 2006
355
Estimating Tail DevelopmentFactors
Link
Ratio
Relafivi~,of
Triangle
Estimated Benchmark Development
Months of
bv
Link
to
Matufi~"
Triangle
Ratio
Benchmark
12
Z00(
2.00(
100%
24
1.45(
1.350
129%
3{
1.20(
1.150!
133%
48
1.15(
1.10(
150%
6(]
72
84
9{
108
1.10(
1.08(
1.05(
1.03~
1.01(
Tail
1.051
1.03C
1.025
1.02(
1.01C
100%
1.05(
Ehosen Ratio
;Implied Tail
200%
267°/,
200°A
175°A
175%
]
1.08[
In the case above, we judgmentaUy select that the triangle development is roughly 175% of
benchmark based on the 60 through 108 m o n t h relativities. So the .05 development portion
of the benchmark tail becomes .05x1.75=.0875~.088. Consequently the entire tail factor,
including tmity, is 1.088.
4.2.2 A n o t h e r i m p o r t a n t n o t e
It is important to consider that adjusting the benchmark tail for actual triangle link ratios is
only helpful as long as the link ratios, or at least the broad pattern of link ratios has statistical
reliability (predictive accuracy). If not, the uncertainty surrounding the true long-term link
ratios of the block of business will cause the adjusted tail factor to lack predictive accuracy.
356
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
4.3
Advantages and Disadvantages o[ Using Benchmark Data
When a good benchmark tail factor is available, this is both one of the easiest and also
among the most usefi.fl methods. However, it is often difficult to find a perfect match in
terms of all the factors (claims handling, case reserving, potential for large claims, etc.) that
affect loss development. Adjusting the benchmark improves the fit markedly. One could
even think of the process of adjusting the benchmark as that of fitting a curve to the link
ratios, where the family of curves you are fitting from consists of various relativity-adjusted
versions of the benchmark. If the benchmark is remotely related to the book of business
being analyzed, that family of ctm'es should be a superior choice to the highly assumptiondriven curve families discussed later under curve fitting.
On the other hand, it is often very difficult to obtain the more-mature data needed to create
a reliable benchmark tail factor. So, for tail factors beginning at 108 or 120 months, it may
be very difficult to find a suitable benchmark.
5. GROUP 4 -THE CURVE HTTING METHODS
As good students of numerical analysis, actuaries long ago realized that they could attempt to
extrapolate the tail development by fitting curves to the development before the tail, then
using the fitted curve to extrapolate the additional tail development. Some methods have
been developed that fit a curve to the paid or incurred loss. Other methods fit to the link
ratios. What they all have in common is that they begin with some assumption about the
development decay that gives rise to a family of curves, and then select the coefficient(s) that
specify the particular member of the family of curves that best fits the data. As with most
extrapolations, they are as good as the assumptions that underlie them.
5.1 McClenahan's Method-Exponential Decay of Paid Loss Itself
McClenahan's method (as discussed in [1]) fits a curve to a set of data per an assumption
that the incremental paid loss of a single accident year will decay exponentially over
increasing maturities of the accident year. In effect, that there was some decay rate 'p' and
that the next month's payout on the accidents in a given month would always be 'p' times
the current month's payments on that given accident month. He combined that with an
assumption that no payments occurr in the first few months of a claim. Putting those pieces
together mathematically, he inferred that the payments in a given incremental month of
maturity (call it 'm') were
Ap(m*)q.
Casualty Actuarial Society Forum, Winter 2006
357
Estimating Tail DevelopmentFactors
In this case A is a constant of proportionality and 'p', (0<p< 1, q= (l-p)) represents the decay
rate ~ and 'a '7 represents the average lag time until claims begin to be paid. A theorem from
the study o f compound interest states that
- ~ A p ~m-~)q= A~'~ p ' q =
m::~
Aq/(1-p)=Aq/q=A.
i=0
So A is actually the ultimate loss for the entire year.
Then, under this assumption, the additional payments or incurrals beyond x months are
theoretically determined by the basic formula, at least once p and a are estimated. A n d there
are several ways to estimate p and a. For convenience, p is monthly, but pl', the annual
decay rate, may be defined as 'r 's. Then r may be estimated by reviewing the ratios of
incremental paid between m + 1 2 and m + 2 4 months to the incremental paid between m and
m + 1 2 months. McClenahan advised that 'a' could be estimated by simply reviewing the
average report lag`) (average date of report-average date of occurrence) for the line of
business.. Then, a curve of the form
Af,
where y is the maturity of the accident year in years before each amount of incremental paid
can be fit to the incremental dollar amounts of paid loss (or incurred loss, as long as no
downward development in incurred loss is present in the development pattern).
Then, McClenahan shows that the percentage remaining unpaid for an entire twelve m o n t h
accident year at m months of (returning to p = r t/12) is
(l_p)X(pm+,-a+ p=+l-a ,+ pm*,+=+...+ p,.+l+n)/(12X(1_p)) = p,~-~-> (1-p'Z)/12q
The tail factor at m months is of course unity divided by the percentage paid at m months,
or
1/(100% - percentage unpaid at m months).
6 McClenahan's model actually incorporates additional variables for trend, etc that may be collapsed into
'p' for purposes of this analysis.
7 In Mclenahan's original paper, 'd' is used instead of 'a'. But, since I have used 'd' to denote the
development portion of the link ratio or development factor, I am using 'a' to denote the average payment
lag.
8 Please note that the usage of 'r' in this context is different than the usage in McClenahan's original paper.
It is used merely because it represents an annual rate.
9 Note that 'a' applies on a month-by-month basis. So it is technically incorrect to say that the average lag
between the beginning of all loss reporting for an accident year is six months (the average lag between
inception of the accident year and loss occurrence, at least for a full twelve month accident year) plus 'a'
months. To simplify the calculations, the first twelve months can be excluded from the fit
358
Casualty Actuarial Society Forum,Winter 2006
Estimating Tail DevelopmentFactors
Substituting our formula for the unpaid at 12 m o n t h s , McClenahan's m e t h o d produces a tail
factor o f
1/{1 - [p"~~'" (1- p ' 3 / 1 2 q ] }
Some algebra reduces that to
1 2 q / { 1 2 q - pm-~-u,(1-
p*-') },
which provides a nice closed form*" expression for the tail.
A n Example:
A s s u m e that you begin with an 8 5 e a r triangle, and generate the following link ratios:
12-24
24-36
,6-48
48-60
;0-72
72-84
84-96
5.772
1.529
1.187
1.085
1.042
1.022
1.012
The first step is to covert t h e m to a form o f dollars paid (remember that there are different
paid amounts for different accident years, so we just begin with one h u n d r e d dollars for the
cun-e fitting and multiply by the successive link ratios.
Development
Stage
12-24
24-36
36-48
48-60
60-72
72-84
~4-96
Equivalent
Link
Beginning
Cumulative
Mamfi~P~d
Ratio
5.772
12
$100.00
1.52c~
24
$577.23
1.18"~
36
$882.45
1.08
4~
$1,047.38
1.047
6C
$1,136.50
1.027
77
$1,184.66
1.01."
84
$1,210.68
9{
$1,224.75
m It should be noted that while a closed form expression makes the calculations easy, for some audiences, it
may be preferable to show the projected link ratios, at least until they are overwhelmingly close to unity.
Casualty Actuarial Society Forum, Winter 2006
359
Estimating Tail DevelopmentFactors
T h e n subtract successive cumulative paid amounts to obtain 'normalized to $100 o f first year
paid' incremental dollars at each stage o f d e v e l o p m e n t that mirror the actual link ratios.
Development
Stage
12-24
24-36
36-48
48-60
60-72
72-84
84-96
Equivalent Incremental
Link
Beginning
Cumulative
Paid
Maturm"
Paid
(Difference)
Ratio
5.772
12
$100.00
$100.0(
1.529
24
$577.23
$477.22
1.187
36
$882.45
$305.2~
1.085
48
$1,047.38
$164.92
1.042
60
$1,136.50
$89.1~
1.022
72
$1,184.66
$48.1(
1.012
84
$1,210.68
$26.0,~
96
$1,224.75!
$14.0e
T h e n ratios o f the successive 'normalized' incremental paid amounts can be taken.
Development
Stage
12-24
24-36
36-48
48-60
60-72
72-84
84-96
Equivalent Incremental
Link
Be~annm~ Cumulanve
Paid
Ratio
Mamrit)Paid
(Difference)
5.772
12
$100.00
$100.0(
24
$577.23!
1.529
$477.22
1.187
36
$882.4"
$305.2,~
1.085
48
$1,047.3~
$164.93
1.042
60
$1,136.5(
$89.1,~
1.022
72
$1,184.66!
$48.1~
84
$1,210.681
$26.0,2
1.012
96
$1,224.75
$14.0{
Year
to Year
Ratio
4.7723
0.6396
0.5404!
0.5404
0.5404
0.5404
0.5404
As one can see, in this contrived example, the d e v e l o p m e n t stage-to-stage ratio is a constant
r = .5404. It's twelve root p is p = r 1/n = .95.
That o f course only provides p, the average delay must be found as well. Because the answer
is contrived to have a=7 m o n t h s , a= 7 m o n t h s will work perfectly n for this example, b u t
note that McClenahan suggests merely using the report delay for the b o o k o f business to
determine 'a'.
Using a= 7 m o n t h s and p = .95, the c o m p u t e d tail factor is
1 2 q / { 1 2 q - .95 m-a-''' (1- .9512)}, = .6/{.6 - .017385(1- .5404)} = 1.0135.
If one reviews the link ratios prior to this, it certainly appears to be reasonable.
extending the payout to additional stages o f d e v e l o p m e n t will confirm its accuracy.
In fact,
n An interested reader can confirm that a=7 months and p=.95 yields the exact link ratios above.
360
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
5.1.1 Advantages and disadvantages of McClenahan's method
At its core, McClenahan's method involves three basic assumptions: First, it assumes that the
pattern of paid loss will be a constantly decreasing pattern, at least after all the initial report
lags are finished. Second, he assumes that the reduction will always occur in proportion to
the size of the most current payout (exponential decay). Third, he assumes that the
exponent of decay is constant throughout the entire payout pattern. Logically speaking, if
one knew nothing about the individual pattern of the data, but was forced to make some
assumptions, those assumptions would seem to be about as minimal and reasonable as
possible (excepting perhaps the third). But it is important to remember that they are
assumptions and as such will color the predictions the method generates. They do suggest
exponential decay of the paid amounts, and exponential decay is a relatively fast decay
relative to other forms of asymptotic (far out in the tail) decay. Moreover, it does seem that
in practice the decay in paid loss often seems to 'stall out' and show less decay near the tail.
5.1.2 I m p r o v e m e n t 2 - exact fitting to the oldest vear
A common problem with fitted curves is that the combination of the curve assumptions and
the data in the middle of the triangle may create a curve that varies significantly from the
development factors at the older stages. McClenahan's method is relatively unique in that
the cutn,e is fit to the incremental paid, rather than the link ratios (as will be done in most of
the later methods). Nevertheless, we can often improve the quality of the tail prediction by
comparing the fitted value to the actual incremental paid loss at the latest stage.
This approach is especially helpful when the curve does not match the shape of the data
itself. For example, assume that the assumption of a constant decay rate does not hold. Say
the initial year-to-year decay was high at between 3612 and 48 months, 48 and 60 months,
etc., but the decay rate at 84 to 96 months and 96 to 108 months, etc. is much less (i.e., a
higher decay factor). Then, the last incremental payments (say between 108 and 120) may be
much higher percentagewise than what is implied by the fitted curve.
In that case 13, one need merely multiply the 'development portion' of the tail factor (the tail
factor minus one) times the ratio of the actual 108 to 120 increment to the fitted increment.
O f course, unit3, (one) must be added to the final result to produce a proper tail factor.
12Note that because of the delay a before payments, etc. begin, the apparent decay between 12 and 24
months and 24 to 36 months is a distortion of the true annual decay.
~3Assuming thatthe data has enough volume for the 108 to 120 link ratio to have full credibility.
Casualty Actuarial Society Forum, Winter 2006
361
Estimating Tail DevelopmentFactors
For example, in the above data, the last incremental data shown is from 96 to 108 months.
In that case the fitted value equals the actual 'normalized' value equals $14.06 per a 96 to 108
link ratio of 1.012 and decay rate of .5404. But what if we had the same decay rate overall,
but the link ratio from 96 to 108 was 1.018. In that case, the incremental paid would be
$21.09, or 150% of the fitted value of $14.06. Then the adjusted tail factor would be:
l+150%(fitted tail factor-I) = 1+150%(1.0135-1)=1+150%x.0135=1.0203.
Note that in the case of McClenahan's method, the ratio used for 'exact fitting' is the ratio of
actual to fitted paid loss. In the later methods, where a curve is fit to the 'development
portions', a ratio of development portions should be used to produce the exact fit to the last
link ratio.
5.1.3 I m p r o v e m e n t 1 (using multiple years to estimate the tail) can enhance
improvement 2
For McClenahan's method, and all the curve-fitting methods, improvement 1 can only be
done in connection with improvement 2. In essence, the concept is to create an exact fit to
the next-to-oldest link ratio or 'normalized' paid loss, and perhaps the third-to-last link ratio
as well. Then, the implied tail factors can be averaged or otherwise combined into a single
tail factor indication. This method is particularly useful when the 'tail' of the triangle has
some credibility, but the individual link ratio estimates from the development triangle are not
fully credible.
Dev
Sta~e
12-24
24-36
Link
Eqmvalent
Cumulative
Endin F
Ratio
~htufiw
5.772!
12
1.529
24
Incremental
Paid
Prod
~Differenc~
$100.00'
$100.00
Year
to Year
Rauo
Rexfised Equivalent Incremental
Link
Cumulative
Pmd
Ratio
Pmd
(Differenc~
5.772
$100.00
$100.00
4.7723
1.529
$577.23
$477.23
0.6396
1.187
$882.45
$305.22
0.5404
1.085 $1,047.38
$164.93
36-48
¢8-60
1.187
1.0851
36
48
$577.23
$882.45
$1,047.38
$477.23
$305.22
$164.93
50-72
1.0421
6G
$1,136.50!
$89.12
0.5404
1.042
$1,136.50
$89.12
72-84
84-96
1.02~
1.012
72
84
$1,184.66]
$1,210.68
$48.16
$26.02
0.5404
0.5404
1.044
1.018
$1,184.66
$1,236.79
$48.16
$52.13
96
$1,224.75
$14.06
0.5404
$1,259.05
$22.26
For example, the table above contains the data cited in the original example of McClenahan's
Method (5.1) as the first set of link ratios, equivalent cumulative paid, etc. But, beginning
with the 'Revised lank Ratio' column it contains alternate link ratios, etc. for 72 months and
later. Using that data, one would still conclude that the fitted annual decline is.5404. But,
now the last rink is 1.018 (as in 5.1.2 - Improvement 2) instead of 1.012, and that the nextto-last (penultimate) 72-84 rink is 1.044 instead of 1.022. In this case, the implied normalized
incremental paid between 72 and 84 monthsis now $52.13 instead of the original $26.02.
$52.13 is approximately twice $26.02, so the 72-84 activity would imply a tail factor of
362
Casualty Actuarial Society Forum,Winter 2006
Estimating Tail DevelopmentFactors
l+200%(fitted tail f a c t o r - I ) = 1+200%(1.0135-1) = 1+200%x.0135 = 1.0270.
The implied tail factor per the 84-96 link ratio is very close to the 1.0203 of the previous
example. Note that the normalized paid loss in the 84-96 stage is $22.26 now or roughly
158% of paid loss. That implies a tail factor of
1+158%(1.0135-1) =1+158%x.0135 = 1.0213.
So, averaging the two, a tail factor in the range of 1.024 might be optimal.
5.2 Skurnick's 14Simplification of McClenahan's Method
Skumick's approach in [3] is essentially the same as McClenahan's. The difference is that
Skumick does not include the delay constant. Further, Skumick does not calculate a single
decay rate for the entire triangle using selected link ratios. Rather Skumick fits a curve to
each accident year and uses each cuta-e as the sole mechanism of projecting each year's
ultimate losses. MathematicaUy, his tail factor reduces to
(1-~(l_r_
ry )
where r and y are as before. In this case y denotes the number o f years o f development at
which the tail factor will apply.
An E:eample
Consider the following incremental loss payouts:
Development
Stage
12
24
36
4~
6G
72
84
96
Accident Year
199i
1991
400(
2000
100(
500,
25(
12~
62.5
1000
2000
1000
500
250
125
62.5
31.25
14This method is also referred to as the 'Geometric Curve' method.
Casualty Actuarial Society Forum, Winter 2006
363
Estimating Tail DevelopmentFactors
F o r illustration o f the curve fitting process, the 1992 data p r o d u c e s the following table, w h e n
a curve is fit to the natural logarithms o f the paid loss in each year (using the identity
~ ( a x ~ ) = ~(.a)+yXk,(r) ).
Development
Stase
Stage
in Years
Amount Log of
Paid
~mount
Fitted Line
Ln(A) = 8.987 EXP = A =
Ln(r) =
-.693
EXP = r =
8000
Fitted
Fit
0.5
Cu~'e
gr~or
12
24
36
1
2
3
4,00(
2,00(
1,00(
8.29405
7.600902 !
6.907755
4,000
2,000
1,000
48
60
72
84
4
5
6
7
50C
25(
125
63
6.21460{
5.521461!
4.82831~
4.135161
500
250
125
63
T h e tail factor is t h e n (1-.5)/(1-.5-.57)=.5/(1-.5-.007813) = 1.0159.
T h e above is o f course a contrived example. B u t consider the m o r e typical case o f the 1991
accident year. In this case, the p a y m e n t s begin low, t h e n decrease after reaching a ' h u m p ' in
t h e 24 m o n t h stage. T h e eventual rate o f decrease is still .5, b u t the curve fit produces:
Fitted Line
Development
Stage
Amount
Stage
m Years
Paid
12
24
36
48
60
72
84
96
1
2
3
4
5
6
7
8
1,00C
2,00C
1,00¢
50C
25C
125
63
31
Los of
Ln(A) =
8.294 EXP = A =
4000
Fitted
F~t
kmount
Ln(r) =
-0.578 EXP = r =
0.56123
Cur~,e
Error
6.907755
7.60090;
6.907755
6.21460{
5.521461
4.82831~
4.13516~
3.44201~
2,245
1,122
561
281
140
70
35
18
Because o f the h u m p shape 'r' is c o m p u t e d at a higher (i.e., less decay) value, .5613. H e n c e
the tail factor is m u c h larger at
(1 -.5613) / (1 -.5613-.56137) = .4387/(.4387-.017554) = 1.041 7.
364
Casualty Actuarial Society Forum, Winter 2006
-1,245
878
439
219
110
55
27
14
Estimating Tail DevelopmentFactors
5.2.1 A d v a n t a g e s a n d d i s a d v a n t a g e s of Skurniclds m e t h o d
The primary advantage of Skumick's method, at least relative to McClenahan's method, is
that the calculations are much simpler. But correspondingly, this method invoh'es not only
all of the assumptions underlying McClenahan's method; a constantly decreasing pattern,
exponential decay, and a lack of trend in the decay rate; it adds the assumption of no lag
between the accident date and when payments begin. The last assumption is clearly untrue
in the vast majority of cases.
As shown above, an additional major disadvantage is that it does not accommodate 'hump
shaped' patterns well. The problems with hump-shaped curves sen-e as an introduction to
the next improvement.
5.2.2 I m p r o v e m e n t 3 - limit curve f i r i n g to t h e m o r e m a t u r e years
Skurmck's method is a prime candidate for this approach, because it is so c o m m o n to have a
'hump-shaped' payout curt-e, whereas by the very nature of the exponential curve,
exponential curves are monotonically decreasing. So, it is logical to refocus the tail
estimation process, putting primary emphasis on the tTpe of claims activity occurring near
the tail.
Going back to the 'Brief Digression' on types of claims activity, the t3"pe of claims activity
most closely associated with the tail does not begin until after 48 or 60 months. So, it would
be logical to just fit the development cun, e to the paid after 60 months. The result of
performing that limited fit on the 1991 data used to illustrate Skumick's method is shown
below.
Development
Stage
Stage
in Years
72
84
96
Amount Log of
Paid
6
7
8
Amount
12~
62
31
Freed Line:
Ln(A) = 8.987 EXP = A =
Ln(r) =
-0.69 EXP = r =
4.82831
4.13519
3.4420~
8000
Fitted
Fit
0.5
Cun,e
Error
125
62
31
As expected, this produces the correct decay rate value of 'r' = .5, and the corresponding tail
factor of 1.0159.
Casualty Actuarial Society Forum, Winter 2006
365
Estimating Tail DevelopmentFactors
5,2.1 A note of caution
The above improvement is logical and generally works well with large volume highcredibility data. When the mangle is of 'medium 'is size and has a fairly high cap on loss size,
the mangle will not have full credibility. Therefore, a fit to paid data directly out of the
mangle will likely lead to poor tail factor estimates. O f note, Skumick's method is not the
only method where this will yield poor tail esnmates. It will happen with all the curve-fitting
methods.
5.2.3. Improvements 1 and 2 applied to Skumick's method
These improvements and their processes have likely been discussed enough earlier in this
paper to eliminate a need for examples. Logically, both improvements may be applied while
using Skurnick's method.
Method 1, using multiple ending years can be applied by simply fitting the curve to all the
payments but the last year, computing the corresponding tail factor for the next-to-last stage
of development, and dividing by the last link ratio.
Method 2 can be performed just as it was in McClenahan's method. For example, in the
poor curve fit obtained when fitting to all of the 1991 data, the 'development portion' of the
fitted tail, 1.0417-1 =.0417 could be multiplied by the ratio of the actual incremental paid loss
in the 96-108 stage (31, holding the place of the exact value 31.25) to the fitted value
(rounded to 18). Note though, that the 'corrected' tail factor is even further off at
1 + 31 ×.417 / 18 = 1.0718. This illustration of when improvement 2 does not improve the tail
factor prediction is intended to further show what happens when the t3.'pe of curve fitted is a
poor match for the pattern of the data.
5.3 Exponential Decay of the Development Portion of the Link Ratios 16
This method is the first of several methods that extrapolate the tail factor off the loss
development link ratios rather than the paid loss. This method was referred to briefly in the
discussion of the Bondy method as a possible source of theoretical underpinnings for the
two Bondy methods. The process is very simple. Given a set of link ratios l+dl, l+d2,
l+d3,.... 1+~., a curve of the form
Dxr
=
where D is the fitted development portion of the first link ratio and r is the decay constant,
is fit to the dm's. The easiest way to do so is by using a regression to the natural logarithms
of the dm's. Then, for an ending dy of small size, the additional development can be
estimated by using the previous approach of
LsIt is very difficult to quatify 'medium' in a manner that will work across the different lines of insurance
and still be meaningful years in the future. At the time this was written, an example of a 'medium" volume
triangle might be a very large workers compensation self-insurance fund.
16This method was outlined in Sherman's paper, but likely was already heavily used by actuaries before
Sherman's paper was published..
366
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
f i (1+ D x r ' )
hi=) +l
=[I(l+d:.r"')
m=l
= l+d,.
~
'
rm = l + d ~ z / ( 1 - r ) .
m=3+l
This also automatically introduces Improvement 2 by fitting exactly to the last point. Similar
algebra would show that the tail factor is approximated by
1+ D x r y+I /(l-r).
For an ending dy of larger size, it may be necessary to simply project the link ratios for the
next fifteen or so years (until the additional tail is immaterial), then multiply them all together
to create a tail factor.
5.3.1 A n e x a m p l e
Consider the following sample link ratio data.
Development
Sta[{e
StaRe
m Years
Link
Rauo
1
2
3
4
5
6
7
2~
3{
4{
6(
72
84
1.5
1.25
1.125
1.0625
1.03125
1.015625
1.007813
The astute reader will notice that is a pattern similar to that underlying the Bondy method.
In any event, to fit our exponential curve to the development portion, we first subtract unity
to obtain the development portion of each link ratio.
Development
Stase
12
24
36
48
60
72
84
Sta~e
in Years
Link
Ratio
1
2
3
4
5
6
7
1.5
1.25
1.125
1.0625
1.03125
1.01562~
1.007812
Development
Portion 'd'
0.5
0.25
0.125
0.0625
0.03125
0.015625
0.0078125
Casualty Actuarial Society Forum, Winter 2006
367
Estimating Tail DevelopmentFactors
Then, as a precursor to c u n ' e fitting, we take the natural logarithms o f the development
portions, or "d's".
Development
Stage
Stage
Link
Development LoB of
in Years R a t i o
Portion 'd'
d'
1~
1
1.5
0.5
-0.69315
2a
2
1.25
0.25
-1.38629
-2.07944
3{
3
1.125
0.125
4{
4
1.0625
0.0625
-2.77259
6(
5
1.03125
0.03125
-3.46574
7~
6
1.015625
0.015625 -4.15888
84
7
1.007813
0.0078125 -4.85203
Then, we fit a line to those logarithms. Standard commercial spreadsheet software produces:
Development
Stage
Stage
Link
in Years R a t i o
Development
Log of
Fitted Curve Values
Portion 'd'
d'
Slope
Intercept
368
1~
1
1.5
0.5
-0.69315
24
2
3{
3
1.25
0.25
-1.38629
1.125
0.125
4~
6£
4
5
1.0625
1.03125
-2.079&
0.0625
0.03125
-2.77259
-3.46574
72
84
6
7
1.015625
1.007813
-0.6931
0.0000
0.015625 -4.15888
0.0078125 -4.85203
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
T h e n , o u r ' D ' , or d e v e l o p m e n t p o r t i o n at time zero, is the e x p o n e n t o f the intercept, a n d the
rate o f reduction, 'r' is the e x p o n e n t o f the slope. Calculating t h e e x p o n e n t s a n d the fitted
curve, we get:
Development Stage
Sta~e
m Years
12
24
36
48
60
72
84
Link
Ratio
1
2
31
4
5
6
7
8
Developmen! Lo~ of Fi,ed Cu~'e Value~
Portion 'd'
d'
5lope
-0.6931
[ntercept
0.0000
1,5
0.~ -0.69315
1.25
0.2~ -1,38629 = exp/slope)
1.125
0.12~ -2.07944D = exp(intercept)
1.0625
0.0625 -2.77259
1.03125
0.03125 -3.46574
1.015625
0.015625 -4.15888
1,007812 0.007812~ -4.85203
9
10
11
12
13
14
15
16
17
18
19
20
21
22
Casualty Actuarial Society Forum, Winter 2006
Fi,ed
Cum-e
0.5
1!
1.50000
1.25000
1.12500
1.06250
1.03125
1.01563
1.00781
1.00391
1.00195
1.00098
1.00049
1.00024
1.00012
1.00006
1.00003
1.00002
1.00001
1.00000
1.00000
1.O000C
1.0000C
1.0000C
369
Estimating Tail DevelopmentFactors
T h e n , for reference we c o m p u t e the tail factor u s i n g b o t h the 'quick' formula usable for
smaU r e m a i n i n g ' d e v e l o p m e n t portions', a n d by multiplying the fifteen fitted link ratios that
m a k e up the tail.
Quick Formula Tail
1+1x(.5^8)/(1-.5) =
1.00781
I
Product of 8-22 Links
1.00783
A s o n e can see, the difference is negligible.
5.3.2 A m o r e r e a l i s t i c e x a m p l e
T h e previous e x a m p l e was contrived to m a k e the m a t h e m a t i c s clear. C o n s i d e r the following
set o f m o r e realistic data.
Development
Sta~e
Link
Rauo
Sta~e
in Years
12
24
36
48
60
72
84
96
1
2
3
4
5
6
7
8
2.000
1.250
1.090
1.050
1.040
1.030
1.028
1.020
A cun-e can be fit to the data u s i n g the m e t h o d o l o g y e m p l o y e d in the previous example.
Development
Stage
Stage
in Years
Link
Ratio
Development
Portion 'd'
Log of
d'
12
24
36
48
60
72
84
96
108
1
2
3
4
5
6
7
8
9
2
1.25
1.09
1.05
1.04
1.03
1.028
1.02
1.018
1
0.25
0.09
0.05
0.04
0.03
0.028
0.02
0.018
0.0000
-1.3863
-2.4079
-2.9957
-3.2189
-3.5066
-3.5756
-3.9120
-4.0174
Fitted Cun-e Values
Slope
-0.4415
Intercept -0.5723
r = exp(slop~
D = exp~ntercepO
]
0.643042
0.56422
Fitted
Curve
Fit
Error
1.3628
1.2333
1.1500
1.0965
1.0620
1.0399
1.0257
1.0165
1.0106
-0.6372
-0.0167
0.0600
0.0465
0.0220
0.0099
-0.0023
-0.0035
-0.0074
N o t e that the fit errors exhibit s o m e c)clic behavior, negative as a g r o u p at first, t h e n
positive f r o m 3-6 years, t h e n negative again at 7-9 year maturities. This suggests that the
370
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
curve may be a p o o r fit. T h a t is b o r n e out by the relationship o f the tail factor estimates
with and w i t h o u t exact fit to the last rink ratio:
( • u iFormula
c k Tail
1,,+,Dx(r^lO//(1-r ) =
1.019108
Product of 8-22 Est. Links
1.019226
I
After exact fit to last link
1+.0191 x.018/.01061
]
1.032403
O n c e again the 'quick approximation' to the tail is almost identical to the precise tail
indicated by exponential decay. However, note that because o f the p o o r fit o f the curve near
the tail, the use o f I m p r o v e m e n t 2 (exact fitting to the last link ratio) produces a markedly
different tail factor. T h e question o f w h i c h tail factor is best m u s t n o w be answered.
T o do so, I m p r o v e m e n t 3 (fitting the curve solely to the mature years) is in order. In this
case, the curve will simply be fit to years 4 (48 m o n t h s ) and beyond. T h a t produces the
following fit;
Development
Stase
Sta~e
inYears
48
60
72
84
96
108
4
5
6
7
8
9
Link Development
Ratio
Portion 'd'
Lo~ of
d'
1.05
1.04
1.03
1.028
1.02
1.018
-2.9957
-3.2189 r = exp(slope)
-3.5066 D = exp0ntercepO
-3.5756
-3.9120
-4.0174
0.05
0.04
0.03
0.028
0.02
0.018
Fitted Cun,e Values
Slope
-0.2073
Intercept -2.1900
0.812748
0.111915
Fitted
Curve
Fit
Error
1.0488
1.0397
1.0323
1.0262
1.0213
1.0173
-0.0012
-0.0003
0.0023
-0.0018
0.0013
-0.0007
W h i c h produces the following tail estimates:
Quick Formula Tail
l+Dx(r^l~)/(1-r/=
1.075166
Product of 10-24 Est.
Links
1.075813
I
After exact fit to last link
1+.075x.018/.0173
I 1.078035
Casualty Actuarial Society Forum, Winter 2006
371
Estimating Tail DevelopmentFactors
Due to the low fit errors, as long as the 48-120 development triangle data that generated the
link ratios is credible, this would strongly suggest that a tail factor of around 1.075 is needed.
Note also that the 'quick approximation also works well in this instance. In summary, this
example illustrates the importance of restricting use of the fitted curve to the portion of the
development data that it can reasonably fit.
5.3.3 Advantages
and disadvantages of this method
v
A primary advantage of this method is it's simplicity. The assumption of exponential decay
is relatively easy to understand. The calculations have moderate complexity, but an
illustration of the fitted values can readily give laypeople comfort that the method is being
executed correcdy. O f note, this method is 'asymptotically equal' to both McCleuahan's and
Skumick's methods, yet is much simpler to execute. That also leads to it's major
disadvantage. Because it assumes such a quick decay of the [ink ratios (exponential decay is
faster decay than l / x , l / x 2, l / x 3, etc.), it can easily underestimate the tail.
5.4 Sberman's Method - Fitting an Inverse Power Curve to tbe Link
Ratios
This method, the l a s t 17 of the curve fitting approaches to be discussed, was first articulated
by Sherman [2]. Sherman noted TM, while fitting a curve from the McCleuahan-SkumickExponential Decay family, that the 'decay ratios' (ratios of successive development portions
of link ratios) were not constant as suggested by expoential decay. Rather, as one went
further out in the development pattern, the decay ratios rose towards unity (i.e. there was
less and less decay as one went further out in the curve). Looking at the data, it appeared
that asymptotically, the decay ratios approached unity. Based on this, he posited an 'inverse
power' curve of the form l+aXt b (t representing the maturity in years) for the link ratios.
Sherman then investigated the quality of curve fit to actual industry data for several families
of cmn-es, including the inverse power culn-e. The family that he found generally fit best
were the so-caned 'inverse power' culx-es.
The process of fitting an inverse power curve is very similar to that used to fit the
exponential ctuve, excepting that the 'independent variable' used in the curve fit is In(t).
More specifically, the identity
In(l+d-1) = In(d) = In(l+aXtb-1) = In(axt b) = In(a) + bxln(t)
can be used to create an opportunity to base the fitted curve on a simple regression.
17 Sherman also discussed the fitting of a lognormal curve to the cumulative paid (or implied cumulative
paid) and the fit of a logarithmic curve to the link ratios. However, the lognormal fit does not lend itself to
easy spreadsheet mathematics, and the logarithmic fit to the link ratios does not produce a unique tail
factor. Further, a Sherman discussed, the inverse power curve is a preferable approach.
~8Mr. Sherman discusses this in Section III of [3].
372
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
Unfortunately, this author is n o t aware o f any simple closed form approximation to the tail
this curve generates, so the tail factor m u s t be estimated by multiplying together the
successive link ratios after the tail begins until the impact o f additional link ratios is
neg~gible.
5.4.1 A n e x a m p l e
This may best be illustrated by using the initial dataset used for the exponential decay
approach:
Development
Stage
Stage
in Years
Link
Ratio
12
1
1.5
24
2
1.25
36
3
1.125
48
4
1.0625
60
1.03125
72
~ 1.015625
84
1.007813
The first step is to calculate the d e v e l o p m e n t portion o f each link ratio and take natural
logarithms o f the result.
Development Sta~e
Sta~e
in Years
12
24
36
48
60
72
84
Link
Ratio
4
5
6
7
1.5
1.25
1.125
1.0625
1.03125
1.015625
1.007813
Development
Portion 'd'
0.5
0.25
0.121
0.0625
0.03125
0.01562.
0.0078125
Log of
d'
-0.6931
-1.3863
-2.0794
-2.7726
-3.4657
-4.1589
-4.8520
Casualty Actuarial Society Forum, Winter 2006
373
Estimating Tail DevelopmentFactors
T h o s e will r e p r e s e n t t h e ' d e p e n d e n t variable' in o u r regression. T h e n for t h e i n d e p e n d e n t
variable, we take natural logarithms o f the d e v e l o p m e n t s t a g e / b e g i n n i n g maturity for the
link ratio in years.
Development
Stage
Stage
in Years
Link
Ratio
Development
Portion 'd'
1.5
1.25
1.125
1.0625
1.03125
1.015625
1.007813
24
3C
4{
6(
7]
84
0.5
0.25
0.125
0.0625
0.03125
0,015625
0.0078125
Lo~ of
Log of
d'
Stage in Yrs
'X'
W'
-0.6931
0.0000
-1.3863
0£931
-2.0794
1,0986
-2.7726
1,3863
-3.4657
1,6094
-4.1589
1.7918
-4.8520
1.9459
T h e n , we c o m p u t e the regression parameters.
Development
Stage
12
24
36
48
6C
72
84
374
Stage
in Years
Link
Ratio
1
2
3
4
5
6
7
Development
Port/on 'd'
12
1.25
1.12~
1.062~
1.0312~
1.01562~
1.007812
Log of
Losof
Fitted Curve Parameters
d'
Stage inYrs
LX'
'3"
31ope =
-2.1051~ =b
0.5
-0.6931
0.000£ [ntercept =
-0.20881
0.25
-1.38631
0.811553
0.6931 = exp(intercp9
0.125
-2.0794
1.0986
0.0625
-2.772(
1.3863
0.03125
-3.4651
1.6094
0.015625] -4.158S
1.7918
0.0078125
-4.852(
1.9459
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
Following that, we c o m p u t e the fitted curve values a n d the fit error.
Development
Sta~e
Stage
in Years
Link
Ratio
12
24
1
2
36
48
60
72
84
3
4
5
6
7
Fitted Curve Parameters
Slope =
1.5 Intercept =
1.25 a = exp(intercept)
-2.10512 =b
-0.20881
0.811553
1.125
1.0625
1.03125
1.015625
1.007813
8
9
10
11
12
13
14
15
16
17
lg
2£
21
2~
Fitted
Zurve
Fit
Error
1.8116
1.1886
1.0803
1.0438
0.3116
-0.061z
-0.0447
-0.0187
1.0274
1.0187
1.0135
1.0102
1.008C
1.0064
1.005~
1.0042
1.0037
1.0031
1.0027
1.0024
1.0021
-0.0038
0.0030
0.0057
1.0018
1.0016
1.0015
1.0013
1.0012
A n d , the taft factor estimates are:
Fitted Tail =
1.056977
Exact Fit to last link
1+0.056977 x0.007813/0.0135
=
I 1.032975[.
E v e n with the utility this adds m the fit, the initial fit p r o d u c e s a tail factor o f over 1.05,
w h e n the previous exponential decay analysis s u g g e s t e d only 1.00781. T h e exact fit
correction, t h o u g h , does p r o d u c e a n u m b e r that is m u c h closer to the theoretical tail.
Casualty Actuarial Society Forum,Winter 2006
375
Estimating Tail DevelopmentFactors
Again, o n e a p p r o a c h is to fit solely to the m a t u r e years.
following regression calculations:
Development StaRe
, Stage
in Years
48
60
72
84
Link
Ratio
Developmenl Los of
Portion 'd'
d'
~'X'
4
1.0625
5 1.03125
6 1.015625
~ 1.007813
0.0625
0.03125
0.015625
0.0078125
-2.7726
-3.4657
-4.1589
-4.8520
That approach produces the
Lol~ of
Fitted Curve Parameters
Stage in Yrs
W'
Slope =
-3.69867 =b
1.3863 Intercept =
2.413854
1.6094 a = exp(intercpt) 11.17696
1.7918
1.9459
A n d t h e n it p r o d u c e s the following fitted ctm-e:
Development! Sta~e
in Years
• Stage
Link
Ratio
Fitted Curve Parameters
~kll~re
Slope =
4{
6(
7]
84
4
5
6
7
Fitted
1.0625Intercept =
1.03125a = exp(intercpO
1.015625
1.007813
Fit
Error
-3.69867 =b
2.413854
11.17696
1.0663 0.0038
1.0290 -0.0022
8
9
10
11
12
13
14
15
16
17
1.0148 -0.0008
1.0084 0.0006
1.0051
1.0033
1.0022
1.0016
1.0011
1.0008
1.0006
1.0005
1.0004
1.0003
18
19
2C
21
1.0003
1.0002
1.0002
1.0001
22
1.0001
A n d , the tail it produces, a l t h o u g h it r e m a i n s higher t h a n the theoretical tail (at a certain
level, the slower decay o f the inverse p o w e r curve as c o m p a r e d to an e x p o n e n t i a l curve
m a k e s it inevitable that it will p r o d u c e a higher tail) is m u c h closer to the theoretical tail.
Fixed Taft=
1.017077
Exact Fit to last link
1+0.017077 x0.007813/0.0084
=
[1.0158841
376
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
5.4.2 T h e more realistic e x a m p l e
G o i n g back to the exponential decay, a tail was fit to the m o r e realistic link ratios s h o w n
below:
Developmenl
StaRe
in Years
Stage
12
24
36
48
60
72
84
96
108
Link
Ratio
1
2
3
4
5
6
7
8
9
2
1.25
1.09
1.05
1.04
1.03
1.028
1.02
1.018
A s in the previous example, we fit an inverse p o w e r curve:
Development Stage
Sta~e
in Years
12
24
36
48
6C
7,7
84
9~
10~
Link
Ratio
~
4
5
6
7
8
91
2
1.25
1.09
1.05
1.04
1.03
1.021
1.02
1.018
Development
Portion 'd'
0.25
0.0c,
0.05
0.04
0.02
0.02~
0.0,~
0.01~
Log of
Lo[[ of
Fitted Curve Parameters
d'
Stase in Yrs
-1.82497, =b
'X'
W'
Slope =
0.0000
0.000C Intercept =
-0.18424
-1.3863:
0.6931 = exp(intercpt)
0.83174
-2.40791
1.098~
-2.99571
1.3862
-3.2189
1.6094
-3.5066
1.791~
-3.5756
1.945~
-3.9120
2.0794
-4.017z
2.197~
A n d t h e n we c o m p u t e the fitted curve values for the link ratios that c o m p r i s e the tail. Since
the link ratios decay so slowly, we project thirty years o f additional d e v e l o p m e n t instead o f
fifteen.
Casualty Actuarial Society Forum, Winter 2006
377
Estimating Tai/ DevelopmentFactors
Development
Stage
Link
Stage
in Years
Rado
Fined Curve Parameters
I
,3lope =
1
2.00C,!Intercept =
24
2
1.25C a = exp(mtercpt)
3~
3
1.0%
4~
4
1.05C
6£
5
1.04C
72
6
1.03C
84
7
1.028
96
8
10~
9
10
378
I
I
Fitted
Fit
Cu~re
Error
-1.82492 =b
-0.18424
1.8317 -0.168.~
0.83174
1.2348 -0.015;
1.1120 0.022(
1.0662
0.0162
1.0441
0.0041
1.031(
0.001(
1.014~
1.023~
1.02G
1.0111
1.018~
1.018
1.0151 -0.002~
i
i
I
I
i
1.0124
11
1.010~
12
1.008~
13
1.007;
14
1.006~
15
1.005~
16
1.0052
17
1.0047
18
1.0042
19
1.003~
20
1.0035
21
1.0032
22
1.003C
23
1.0027
24,
1.0025
2.:
1.0023
2C
1.0022
2~
1.002G
2{
1.0019
2~
1.0018
3(
1.0017
31
1.0016
3~
1.0015
32
1.0014
34
1.0013
3.=
1.0013
3(
1.0012
37
1.0011
3{
1.0011
3c~
1.0010
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
T h a t p r o d u c e s the following tail data.
Fitted Tail =
I
1.114487
I
=
I
1.136502
F o r c o m p a r i s o n , the final ' b e s t estimates' u s i n g the exponential decay were in the 1.03-1.05
range. But, t h o s e best estimates were based o f f a fit to just the m a t u r e years. So, let us fit
the curve solely to the 4 8 + m o n t h data.
Development Stage
Stage
m Years
48
60
72
84
96
108
Link
Ratio
Development Log of
Portion 'd'
d'
'X'
Log of
Fitted Cuta'e Parameters
Stage m Yrs
W'
Slope =
-1.2810~ =b
41
5
1.051
1.04
0.05
0.04
-2.9957
-3.2189
-1.1868~
1.3863 Intercept =
1.6094 a - exp(intercpt / 0.305171
6
7
8
9
1.031
1.028,
1.01
1.018,
0.03
0.028
0.02
0.018
-3.5066
-3.5756
-3.912C
-4.0174
1.7918
1.9459
2.0794
2.1972
H o w e v e r , in this case, the tail is even higher, per the fit
Development Stage
Stage
mYears
48
60
72
84
96
108
4
5
6
7
8
9
10
11
12
13
Etc.
Link
Ratio
Fitted Cuta,e Parameters
Slope =
1.05 Intercept =
1.04 a = exp(mtercpt)
1.03
1.028
1.02
1.018
Casualty Actuarial Society Forum, Winter 2006
-1.28108 =b
-1.18688
0.305171
Fitted
Fit
Curve
Error
1.0515 0.0015
1.038~! -0.0011
1.0307 0.0005
1.0252 -0.002~
1.0213 0.0013
1.0183 0.0003
1.016C
1.0141
1.0126
1.0114
Etc
379
Estimating Tail DevelopmentFactors
Multiplying the link ratios that comprise the tail factor together, the estimated tail is:
Fitted. Tail =
1.208566
Exact Fit to last link
1+0.2086 x0.018/0.0183
]
1.20518
So, this illustrates how this method is generally more conservative than the exponential
decay method.
5.4.3 Advantages and disadvantages of Sherman's method
Relative to the other crux-e-fitting methods, this method's primary strengths and weaknesses
stem from it's source, although that is mitigated by the fact that in choosing the form of the
mathematical curiae family that was used (the inverse power cuB-e), Sherman relied heavily
on actual data. Specifically, he noted that exponential decay factors flattened heavily (i.e.,
rose toward unity) at later ages. So, he chose the inverse power curve as his model to reduce
the decay at later ages. In a sense, Sherman designed the inverse power curve with an eye
toward mathematically correcting an observed deficiency in the exponential decay method.
The approach he used to correct exponential decay 19was merely to find a curve that roughly
matched the data he obsein-ed. So, since the inverse power approach is based on actual
properties of the observed development link ratio curves, and appears to have superior fit to
the data, it should arguably be a better predictor of the tail. But on the other hand it also
gives no single simple assumption (such as decay proportional to development portion size)
that we can test the data against. In other areas, the fit looks a little more mathematically
complex to the outsider, but is no more computationally difficult for the practitioner than
exponential decay of the link ratios.
5. 5 Sherman's Revised Method - Adding Lag to the Inverse Power Curve
In his study of the inverse power curve, Sherman [3] noted that the fit could sometimes be
improved by adding a lag parameter to the curve. He used the formula
l + d ~ l+aX(t-c) b.
In this case, the mechanics of fitting the curve are somewhat more complex. An example
will illustrate the process.
19Sherman effectivelyreplaced 1+ D r' from exponential decaywith l+axta' . Note that a in the inverse power
curve plays the same role as D in exponentialdecay, so reallyhe just replaced r t , wath a constant decay ratio of
r by tb'with a decay rate of
380
((t + l)+ t) b , which is asymptoticallyone.
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
5.5.1 E x a m p l e
of fitting an inverse power curve with lag
W e first set the lag equal to o n e (unity) to b e g i n the p r o c e s s , t h e n fit the an i n v e r s e p o w e r
curve reflecting that lag
Development Stage
Stage
Link
in Year~
Development Log of
Rauo
Portion 'd'
d'
Stage LogofRev. Fitted Curve Parameters
Minus Lag Stage in Yrs
Lag =
1
3lope =
-1.0273 =b
48
4
1.05
0.05 -2.9957
3.0000
1.098~ Intercept =
60
5
1.04
0.04 -3.2189
4.0000
1.3861
72
84
6
7
1.03
1.02~
0.03 -3.5066
0.028 -3.5756
5.0000
6.0000
1.6094
1.791{
96
8
1.0]
0.02 -3.9120
7.0000
1.9455
108
9
1.01~
0.018 -4.0174
8.0000
2.0794
-1.8324
= exp(intercp~
0.1600
T h e n w e c o m p u t e the link ratios o n the fitted curve, a n d the total s q u a r e d fit error as well
Development
Stage
Link
Stage
in Years
Ratio
Fitted Curve Parameters
Lag =
Slope =
1
Fitted
Fit
Squared
Curve
Error
Error
-1.027387872=b
48
1.05 Intercept =
.1.832444677
1.0385
-0.0115
1.32E-04
6O
1.04 a = exp(intercpt) 0.160021887
1.0306
-0.0094
8.79E-05
72
1.03
1.0254
-0.0046
2.12E-05
84
1.028
1.0217
-0.0063
4.00E-05
96
108
1.02
1.0189
-0.0011
1.22E-06
1.01~
1.0167
-0.0013
1.58E-0~
2.84E-04
W e n o t e that the total fit e r r o r associated w i t h a lag o f o n e is .000284.
Casualty Actuarial Society Forum, Winter 2006
381
Estimating Tail DevelopmentFactors
Next, in order to estimate the optimum lag, we use a bisection process, foUowing the process
above for different potential lags; finding the lowest value of the squared error across a
group of values; and progressively narrowing the range. The computations were as follows,
and only 7 steps were needed. For reference, at each step of the process the lowest value of
the fit error as well as the two adjacent values (the three values generated by the lag points
that will be carried to the next step of the process) are in bold.
gtalge 1
Stage 2
Squared
Error
Lag
-1
(
7.06E-04
1.32E-05
2.84E-04
7.50E-04
1.08E-03
StaRe 3
Lag
-1
-o.5!
o
o.5]
1
Sta~e 5
Sta~e 4
Lag
-0.2~
-0.125
(
0.12~
0.2~
Squared
Error
4.8617VE-05
2.2949E-05
La~
-0.125
-0.06251
0
0.062~
i
1.32454E-05
1.72333E-05
3.28903E-05
0.125
Squared
Error
7.06E-04
1.58E-04
La~
-0.5
-0.25
1.32E-05
9.22E-05
2.84E-04
o
0.25
0.5
Sta[~e 6
Squared
Error
La8
2.2949E-05 - 0 . 0 6 2 5
1.625E-05 -0.03125
1.3245E-05
0
1.366E-05
0.03125
1.7233E-05 0 . 0 6 2 S
Squared
Error
1.58E-04
4.86177E-05
1.32454E-05
3.28903E-05
9.22146E-05
Squared
Error
1.62499E-05
1.43034E-05
1.32454E-05
1.30419E-05
1.36599E-05
Stage 7
Squared
Lag
C
0.015625
0.03125
0.046875
0.0625
Error
Final Selection
1.32454E-05
1.30389E-05
1.30419E-05
1.32502E-05
1.36599E-05
0.02
Note that as the fit error changes little near the minimum point, a rounded value is
acceptable.
382
Casualty Actuarial Society Forum,Winter 2006
Estimating Tail DevelopmentFactors
T h e n , t h a t l a g v a l u e m a y b e u s e d i n t h e f i n a l c u r v e fit.
Development
Stase
Lmk
Fitted Curve Parameters
Sta~e
m Years
Ratio
La~ =
~lope =
48
1.05 [ntercept =
6C
1.04
72
1.02
84
1.02~
9~
= exp(intercpt)
Curve
-1.253797784 =b
-1.23628766
0.290460507
1.0511
1.038~
1.0307
1.0252
1,0f Fitted Tail =
10~
Fitted
0.02
1.230663894
1.01~
1.0214
1.018~
lO~
Exact Fit to last link
11
1+0.2307×0.018/0.0185
1.016~
1.014a
1~"
=
1.012c
1.224464865
13
1.011~
14
1.010{
15i
1.009;
16
1.009(
17
1.008.:
18
1.007,
19
1.007g
20
1.006~
21
1.006~
22
1.006(
23
1.0055
24
1.005,
25
1.0051
26
1.004 c.
27
1.004~
28
1.004
29
1.004
30
1.004~
31
1.003!
32
1.003~
33
1.003(
34
1.003~
35
1.003~
36
1.003:
37
1.003~
38
1.0031
39
1.002!
Casualty Actuarial Society Forum, Winter 2006
383
Estimating Tail DevelopmentFactors
Which provides a shghdy smaller tail.
Fitted Tail =
I 1-2306
5.5.2 Advantages and disadvantages of introducing lae in the inverse power cma-e
Summarizing, we can note that while the lag factor may sometimes mitigate the size of the
tail, the inverse power in general tends to produce a higher tail than the exponential fit.
Although it has not been illustrated herein with actual data, the inverse power curve also
generally indicates higher tail factors than McClenahan's and Skumick's methods, as those
methods tend to produce results that are very similar to that of the exponential decaya'. As
before, the inverse power curve's main attraction is that it simply seems to fit the data better.
However, in introducing lag it is clear that much computational complexity is added. The
practitioner should evaluate whether the additional complexity produces large gains in the
accuracy of the estimated tail factor.
6. SUMMAR Y
Several different methods for assessing tail development were presented, as well as some
refinements. Hopefully, this will help the reader in his or her actuarial practice.
2oThat is because they are simply based on exponential decay of the payments rather than the link ratios. A
little analysis will show that their decay patterns are about equal for 'large' maturities. If in doubt, simply
consider their asymptotic properties.
384
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
Appendix 1-Tail Factor Metbods Based on Counts
A.1 Introduction
Although they are less commonly used, there are several methods for estimating tail factors
that are based on counts. Among these are the Sherman-Diss method, the projected unpaid
severity method, and what is really a older year ultimate loss selection method instead of a
tail factor method for unexpectedly low open counts that is based on maximum possible
costs per claim. All of these methods do have demonstrated limitations, though. So, it is
just as important to understand the limitations of each method as it is to understand the
methods themselves.
A.2 The Sherman-Diss Method
The Sherman-Diss method described in [4] is a specific example of what could become a
class of methods that project the open claim counts at future times, and the cost per claim at
each future period. For the first step, this method involves projecting the likelihood that
each 'mature' (near the tail maturity) workers compensation claim will still be open next year,
the following year, the year following that, etc. using life (mortality) tables and the claimant's
current age. Then, the indemnity (wage replacement) benefits each would receive in each
future period (if they are still alive to collect benefits as estimated using the life table) is
estimated using each worker's current annual benefit, plus an estimate of any inflation in the
benefit (should any be allowed under the law of the injured worker's state). The total
indemnity tail would then be calculated by extending the probability of each claimant's
survival at each future period (the expected open claim counts) times the annual indemnity
benefit. For the medical benefits allowed claimants under the workers compensation laws,
the probability of sun'ival to each future period is extended by the current medical inflated
by an appropriate medical inflation factor. The extension of probability of survival times
medical benefits produce the dollars of medical tail.
A.2.1 Pros a n d c o n s
Due to the complexity of the calculations and the status of this discussion as an appendix
rather than the main paper, an example will not be provided. However, some discussion of
this and the other methods in this appendix is certainly in order.
Casualty Actuarial Society Forum, Winter 2006
385
Estimating Tail DevelopmentFactors
When Sherman and Diss compared their method to other tail factor methods (primarily the
curve-fitting methods) on some specific workers compensation data, they found that it
produced much higher tail factors than the other methods. However, when they tested their
method retrospectively against actual dollar emergence on some Western state fund data,
they found that as claimants achieved advanced ages (roughly at thirty to forty-plus years of
development) the medical became much higher than that predicted by their method. Per
their studies, it appears that as claimants achieve advanced ages, unexpected (at least per life
tables and medical) additional development occurs because the main injury may cause related
illnesses that are exacerbated by age and because family or spousal care for severely injured
claimants must be replaced by nursing home care as the caregivers age and become infirm.
So, at least for direct and unlimited workers compensation benefits, it appears that many
common methods produce an inadequate tail, but that this method does not fully solve the
problem.
Also note that this 'open claim count' method is suitable only for fines where benefits are
paid as long as claims remain open. To this author's knowledge, the only fines of insurance
that have that feature are workers compensation and disability.
Further, this method was designed for direct and unlimited claim costs, when most insurers
purchase some form of specific excess reinsurance that caps the insurer's costs at some 'net
retention'. Note however, that method could be revised by accumulating the total projected
costs paid to each claimant and eliminating the claim once the net retention is reachedZk In
so doing, each claim would be effectively capped at the retention.
Lastly, this method only directly produces a tail factor for the mature years. If there is a low
volume of claims remaining open in the older years (as is often the case), the results of this
method will not be a reliable statistic for projecting the tail on the later years (i.e., they will
lack credibility).
Qualifications aside, this method does create a powerful tool in the right circumstances.
Futher, as time goes by it is possible that other 'remaining open count'-based methods will
be developed.
21 Of note, it is also appropriate to build in any projected costs that exceed the limit of per claim reinsurance
purchased.
386
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
A.3 The Unclosed Count Method
This method also requires qualifications, but is worth discussion. Just as in workers
compensation, the open status of a claim is related to payments. In most other lines the
majority of payments occur at the time of claim closing. So, it is reasonable to suppose that
there would be a method based on the number of claims yet to close and the average cost of
each of those claims. O f course, while it may be relatively easy to estimate the number of
claims that will close in the future as long as the actuary is certain that no further IBNR
claims will materialize; it is usually very difficult if not impossible to estimate the average
costs of closing each claim. However, in some limited circumstances, the average paid loss
per closed claim of the oldest accident year may have reliably and permanently plateaued. In
those specific circumstances (and only those circumstances), it would be appropriate to
multiply the number of unclosed claims by the average paid loss per closed claim from the
latest twelve months for the given accident year.
A.3.1 Pros and c o n s
This method cannot be discussed without discussing the tremendous detraction posed by
blithely assuming that the current average paid loss per closed claim will equal the average
cost of disposing of the open claim inventory. The author has personally seen general
liability data of about 48 months maturity and fairly low volume where the average paid per
claim had leveled off at around $5,000 per claim, where only four claims were open, but they
were all $20,000+ claims. One major problem was that the maturity was only 48 months.
So, the actuary is strongly cautioned to use this only for data of at least 96 months maturity,
preferably 120 months, and to carefully review whether the remaining open claims are of the
same type, class, average demand, etc. as the claims closed between, say, 96 and 120 months.
The actuary is also cautioned that if the data volume is not overwhelming large, the
percentage of claims left open for the older years now may not match the percentage of
claims left open at 120 months or so on the more recent years once they reach the 120
month stage. For example, if only four or so claims are left open on the older years, they
will lack statistical validity (a form of credibility) in predicting what will be open when the
more recent years reach the same development stage. Therefore, they will lack validity in
predicting the tail factors for the more current years.
All that being said, under the right circumstances this can be a useful method. One must
simply make sure that the set of underlying assumptions hold in whatever circumstance the
actuary is using this method.
Casualty Actuarial Society Forum,Winter 2006
387
Estimating Tail DevelopmentFactors
A.4 The Maximum Possible Loss Method
This method is a variant of the unclosed count method. It, however, does not so much
create a taft factor as it does establish a maximum tail for the older years. The core idea of
this method is that, given that the maximum net liability of an insurer is some net retention
'R', the liability for all the open claims should not be more than the sum o f R-paid to date
across all the open claims. So, to use it, given that an accident year is sufficiently mature for
no IBNR claims to be reasonably possible, the remaining amounts to reach the retention (Rpaid to date) are summed across all remaining open claims in the accident year. The result is
not so much an estimate of the tail factor as an upper bound on tail development for that
specific year. So, if application of the tail factor to a given year suggests more development
than is 'possible' per the remaining amounts to reach the retention in the accident year, the
ultimate unpaid loss for that accident year might be capped at the amounts remaining to
reach the retention.
In the (fairly unusual) event that there are enough claims left open for this to be a statistically
valid predictor of the development of the more recent years, it could be used in estimating
the tail factor for all the accident years. But, one would have to be certain that this finding
was statistically consistent with the initial tail factor analysis. For example, if the initial tail
factor came from a curve fitting, it might be reasonable statistically that the curve fitting was
simply using the wrong curve. However, if the initial tail factor came from a 'paid over
disposed' method that also used the actual data in the triangle itself, the tail findings would
suggest the data is intemaUy inconsistent. In that case, greater care must be taken to
understand which method is most accurate for the tail factor to be applied to the more
recent years.
A.4.1 Pros and c o n s
This method improves on the average unpaid loss method by virtue of the fact that the
amount to reach the retention need not be estimated. Rather, it is fact. However, it only
produces an upper bound, not an actual best estimate.
Like the average unpaid loss method, there are often statistical reliability issues when making
inferences about the tail factors of the more recent years. But, one cannot readily dispute
the results as an upper bound for the older years on which the method is apphed, at least as
long as one is certain the prospect of additional IBNR claims is immaterial. So, like the
average unpaid loss method, one must be very careful to make sure the proper assumptions
hold when using it. But, unlike the average unpaid loss method, it has far more certainty
surrounding the loss sizes.
388
Casualty Actuarial Society Forum, Winter 2006
Estimating Tail DevelopmentFactors
Appendix 2-Developing Case Reserves on the Older Years
This method is also not so much a method for estimating the tail factors to use in incurred
or paid loss development as it is a method for estimating ultimate losses in the very mature
years. The scenario this addresses is that of a medium-to-low credibility (medinm-to-low
volume of losses in relation to the net retention) loss triangle. In that scenario it is not
unusual for the remaining unpaid loss in the mature to vaiT significantly depending on
whether a large claim, or a few large claims, or no large claims happen to have occurred and
still be open in the late development stage. In such circumstances, the standard application
of a tail factor may not work simply because there are not enough open claims in the mature
years, or even open claims expected in the tail factor, for the law of large numbers to apply.
In that case, some recognition of the specific cases remaining open (assuming no further
reopenmgs or IBNR claims) will make the resulting ultimate loss predictions for the older
years more accurate.
The process is fairly simple. Given a ratio o f what it actually costs to close cases vs. the case
reserves held from the 'paid loss to reserve disposed of' method, one simply multiplies that
ratio times the case reserve to obtain an estimate of the unpaid loss on each of the very
mature years. The ultimate loss estimate for each of those years would simply be the derived
unpaid loss estimate plus the paid-to-date for each )Tear.
A word of caution is in order, however. Remember that this method was used to estimate
the ultimate loss because the law of large numbers did not work. Therefore, the unpaid
losses derived using this method lack credibility in estimating the tail factor for the less
mature years. So, if this method is used because the remaining unpaid losses are driven by
'luck of the draw '=, it is illogical to use the unpaid losses from this method to estimate a tail
factor for the less mature years.
Pros and cons
This method's inherent advantage is it's usefulness in low credibility situations. It's
disadvantage is that it does not truly produce a tab factor, just some estimates of ultimate
loss for the older years. Further, it assumes no reopenings or true IBNR claims. So, it must
be used with great caution and respect for it's limitations.
22The astute reader will note that the adjusted case reserves are exactly what is used to develop a tail factor
in the 'paid loss to reserve disposed of method. But note that in that instance the tail is presumably based
on case reserves that are large enough to have reasonable credibility.
Casualty Actuarial Society Forum, Winter 2006
389
Estimating Tail DevelopmentFactors
Bibliography
390
[1]
McClenahan, Charles L., 'A Mathematical Model for Loss Reserve Analysis', Proceedingsof the
Casual~,ActuatialSode~,,1975 Vol: L2,2II Page(s): 134-153,
[2]
Skumick, David., Review of 'A Mathematical Model for Loss Reserve Analysis' by Charles
McClenahan, Proceedingsof the CasualD,ActuatialSodeO', 1976 Vol: LXIII Page(s): 125-127
[3]
Sherman, Richard E., 'Extrapolating, Smoothing and Interpolating Development Factors',
Proceedingsof the CasualtyActuarialSocieD,,1984 Vol: LX~'XI Page(s): 122-155
[4]
Sherman, Richard E. and Diss, Gordon F., 'Estimating the Workers Compensation Tail',
Casuals Actuwial Sode~,Forum,Winter 2004 Page(s) 207-282
Casualty Actuarial Society Forum, Winter 2006